The Abel Prize 2018-2022

Preface
Contents
Part I 2018 Robert P. Langlands
Citation
Autobiography
The work of Robert Langlands
Foreword
Contents
1 Group representations and harmonic analysis
2 Eisenstein series
3 L-functions and class field theory
4 Global Functoriality and its implications
5 Local Functoriality and early results
6 Trace formula and first comparison
7 Base change
8 Shimura varieties
9 Motives and Reciprocity
10 The theory of endoscopy
11 Beyond Endoscopy
References
List of Publications for Robert P. Langlands
Curriculum Vitae for Robert Phelan Langlands
Part II 2019 Karen K. Uhlenbeck
Citation
Minimal surfaces and bubbling analysis
Gauge theory and Yang–Mills equations
Integrable systems and harmonic mappings
Mathematical Meanderings
Childhood and Education
Midcareer
Texas and Beyond
A journey through the mathematical world of Karen Uhlenbeck
Contents
1 Introduction
2 Nonlinear systems and p-harmonic functions
2.1 A regularity theorem
2.2 A differential inequality
2.3 Outline of proof of Theorem 2.1
3 Harmonic maps of surfaces
3.1 Background
3.2 Bubbling
3.3 Small energy
3.4 The stress energy tensor and removal of point singularities
4 Harmonic maps in higher dimensions
4.1 Monotonicity of normalised energy
4.2 Minimising maps
4.3 Small energy
4.4 Some further developments
5 Gauge Theory
5.1 Background
5.2 The 1982 papers in Commun. Math. Phys.
5.3 Applications
6 The Yang–Mills equations in higher dimensions
6.1 Hermitian Yang–Mills connections on stable bundles
6.2 Connections with small normalised energy
6.3 Removal of codimension 4 singularities
7 Harmonic maps to Lie groups
7.1 Harmonic maps, flat connections and loop groups
7.2 Uniton addition and instantons on R
7.3 Weak solutions to the harmonic map equation on surfaces
References
List of Publications for Karen K. Uhlenbeck
Curriculum Vitae for Karen Keskulla Uhlenbeck
Part III 2020 Hillel Furstenberg and Grigoriy Margulis
Citation
Autobiography
Autobiography
Appendix 1. Arithmeticity and superrigidity
Arithmeticity of non-uniform lattices
Arithmeticity of uniform lattices
Appendix 2. Homogeneous dynamics and number theory/diophantine approximation
Distribution of values of indefinite quadratic forms at integral points
Diophantine approximation on manifolds
Appendix 3. Fields Medal
Appendix 4. Dissertations
The work of Hillel Furstenberg and its impact on modern mathematics
Contents
Introduction
1 Topological dynamics
2 Stationary dynamical systems and the Poisson–Furstenberg boundary
3 Probability, ergodic theory and fractal geometry
4 Multiple recurrence and applications to combinatorics and number theory
References
The work of G. A. Margulis
Acknowledgments
Contents
1 General introduction
2 Arithmeticity and superrigidity
2.1 Proof of superrigidity
2.2 Proof of arithmeticity
3 Normal subgroup theorem
4 Expanders, relative property (T) and lattices
5 Local rigidity of group actions
6 Dynamical systems on homogeneous spaces: an introduction
7 Quantitative non-divergence
7.1 An elementary non-divergence result
7.2 The general case
7.3 Applications to Diophantine approximation on manifolds
8 Conjectures of Oppenheim and Raghunathan
9 Linearization
9.1 Non-ergodic measures invariant under a unipotent
9.2 The theorem of Dani–Margulis on uniform convergence
10 Partially hyperbolic flows and Diophantine approximation
10.1 Exceptional trajectories
10.2 Cusp excursions
10.3 Effective equidistribution
11 A quantitative version of the Oppenheim Conjecture
11.1 Passage to the space of lattices
11.2 Margulis functions
11.3 A system of inequalities
11.4 Averages over large spheres
11.5 Signatures (2,1) and (2,2)
12 Effective estimates
12.1 Periodic orbits of semisimple groups
12.2 Effective solution of the Oppenheim Conjecture
12.3 Power law estimates in dimension at least
References
List of Publications for Hillel Furstenberg
List of Publications for Grigoriy Margulis
Curriculum Vitae for Hillel Furstenberg
Curriculum Vitae for Grigoriy Aleksandrovich Margulis
Part IV 2021 László Lovász and Avi Wigderson
Citation
Autobiography, mostly mathematical
Meeting with mathematics
Moving around in Hungary and in the world
Tarski’s problem and graph limits
Perfect graphs and combinatorial optimization
Algorithmic geometry and cryptography
Avi Wigderson — a short biography
The Mathematics of László Lovász
Contents
1 Introduction
2 Logic and Universal Algebra – Homomorphisms and Tarski’s Problem
3 Coloring Graphs Constructively (on a Way to Expanders)
4 The Lovász Local Lemma
5 Coloring Graphs via Topology
6 Geometric Graphs and Exterior Algebra
7 Perfect Graphs and Computational Complexity
8 The Shannon Capacity of a Graph and Orthogonal Representations
9 The Ellipsoid Method
10 Oracle-Polynomial Time Algorithms and Convex Bodies
11 Polyhedra, Low Dimensionality, and the LLL Algorithm
12 The LLL Algorithm and its Consequences
13 Cutting Planes and the Solution of Practical Applications
14 Computing Optimal Stable Sets and Colorings in Perfect Graphs
15 Submodular Functions
17 Analysis, Algebra, and Graph Limits
18 Final Remarks
References
On the works of Avi Wigderson
Contents
1 Introduction
2 Cryptography
2.1 Cryptography under computational assumptions
2.1.1 Zero knowledge proofs for all languages in NP
2.1.2 Computationally secure multiparty computation
2.2 Information-Theoretic Cryptography
2.2.1 Multi-Prover Zero-Knowledge Interactive Proofs
2.2.2 Bit commitment scheme in the 2-prover setting
2.3 The Importance of the Multi-Prover Interactive Proof Model
2.3.1 Information-Theoretic Secure Multi-Party Computation
2.3.2 Verifiable Secret Sharing
Gate-by-Gate Emulation in the Malicious Setting
3 Pseudorandomness
3.1 Hardness vs. Randomness
3.1.1 Motivation
3.1.2 Wigderson’s Contributions
3.1.3 Pseudorandom Generators
3.1.4 The Nisan–Wigderson Generator
3.1.5 Pseudorandom Generators from Worst-Case Lower Bounds
3.2 Expanders, Extractors, and Ramsey Graphs
3.2.1 Expander Graphs
3.2.2 Randomness Extractors
3.2.3 Multi-source Extractors and Ramsey Graphs
3.3 Unconditional derandomization
3.3.1 Undirected S-T Connectivity
3.3.2 General Space-Bounded Computation
3.3.3 Constant-depth Circuits and Iterated Restrictions
4 Computational Complexity Lower Bounds
4.1 Boolean Circuit Complexity
4.2 Communication Complexity
4.3 Karchmer–Wigderson Games
4.4 Lower Bounds for the Monotone Depth of ST-Connectivity
4.5 Lower Bounds for the Monotone Depth of Clique and Matching
4.6 The KRW Conjecture
4.7 Communication Complexity of Set-Disjointness
4.8 Quantum versus Classical Communication Complexity
4.9 Partial Derivatives in Arithmetic Circuit Complexity
4.10 Resolution Made Simple
5 Complexity, Optimization, and Symmetries
5.1 Permanent and matrix scaling
5.1.1 Doubly stochastic matrices and their permanents
5.1.2 Matrix scaling
5.2 Noncommutative singularity testing and operator scaling
5.2.1 Completely positive operator and its capacity
5.2.2 Operator scaling
5.2.3 Noncommutative singularity and identity testing
5.2.4 Brascamp–Lieb constants
5.2.5 Polynomial capacity
5.3 Capacity and geodesic convex optimization
5.3.1 The Riemannian geometry of positive definite matrices and geodesic convexity
5.3.2 Geodesic convexity of capacity
5.3.3 Computing the capacity via geodesically convex optimization
5.4 The null-cone problem, invariant theory, and noncommutative optimization
5.4.1 Groups, orbits, and invariants
5.4.2 Capacity and the null cone
5.4.3 Geodesic convexity, moment map, and noncommutative duality
5.4.4 Noncommutative optimization under symmetries
References
List of Publications for László Lovász
List of Publications for Avi Wigderson
Curriculum Vitae for László Lovász
Curriculum Vitae for Avi Wigderson
Part V 2022 Dennis P. Sullivan
Citation
Encounters with Geometry—an Autobiography of Concepts
Dennis Sullivan’s Work on Dynamics
Contents
1 Smooth dynamics
1.1 From topology to dynamics
1.2 Rigidity in smooth dynamics
1.3 Further results
2 Dynamics and ergodic theory of Kleinian groups
The Moebius group
Hyperbolic space
Kleinian groups
Hyperbolic manifolds
Quasi-conformal homeomorphisms
2.1 Sullivan’s rigidity theorem
2.2 Conformal densities and Patterson–Sullivan measures
Conformal densities
Patterson–Sullivan measures: construction
2.3 Further results
3 Holomorphic dynamics
3.1 The reemergence of holomorphic dynamics in Paris in the 1980s
3.2 Conformal measures for rational maps
3.3 The λ-Lemma
3.4 Density of stable maps
3.5 Towards the Fatou conjecture: absence of line fields
3.6 Monotonicity of entropy and the pullback argument
3.7 Renormalisation theory for interval maps
3.8 Real and complex bounds
3.9 Riemann surface laminations and the non-coiling lemma
3.10 Renormalisation theory for circle maps
3.11 The Fatou conjecture in the real setting
3.12 Sullivan’s quasisymmetry rigidity programme
4 Sullivan’s dictionary
5 Final words
References
Sullivan’s Juvenilia: Surgery and Algebraic Topology
Contents
1 Surgery and its classifying spaces
1.1 Further developments
Localization
Bordism and classifying spaces
KO[1/2] orientation
Hauptvermutung and triangulation (and beyond)
Surgery revisited
2 The Adams Conjecture
2.1 Background
2.2 The Statement of the Adams conjecture
2.3 Comments on Sullivan’s proof
2.4 Some of its aftermath
Epilogue: A return to geometry
3 Rational homotopy theory
3.1 Sullivan’s model
3.2 A few words on the proof
3.3 A few more applications
3.3.1 Free loop spaces
3.3.2 The Elliptic/Hyperbolic dichotomy
3.3.3 Quantitative Homotopy Theory
3.3.4 Finite primes and Z
References
List of Publications for Dennis P. Sullivan
Curriculum Vitae for Dennis Parnell Sullivan
Part VI Abel Activities 2018–2022
Photos
The Abel Committee
The Niels Henrik Abel Board
The Abel Lectures
The Abel Laureate Presenters
The Interviews with the Abel Laureates
The Abel Banquet 2003–2022
Addenda, Errata, and Updates1